For each problem:

List all physical model simplifying

assumptions.

Draw a well$-$labeled diagram of the resulting

physical model.

Apply the Laws of Nature to the physical

model and generate the mathematical model

of the physical model.

A high$-$speed solenoid actuator$-$valve system is shown. This type of actuator is

used in hydraulic systems to position spool valves for metering fluid flow.

Electric current flows through the coil of wire that surrounds the plunger and

generates a magnetic field, which produces an attracting force on the plunger.

pulling it to the right. Hence, the electromagnetic force pulls the plunger

toward the center of the coil and closes the air gap. The plunger is connected to

the spool valve via the push rod. The return spring is used to return the

plunger$-$valve mass back to its seated position when the electromagnetic force is

removed,